Math101 · Independent Ontario learning library
Math101learn.math101.caRatios
Ratios compare quantities multiplicatively and power proportions, scale drawings, mixtures, probability, rates, and percent.
A ratio compares two quantities by division. It answers “how many times as much?” or “how much of one quantity for another?”
Reading a ratio
If a box contains $6$ blue tiles and $4$ gold tiles, the blue-to-gold ratio is $6:4$, “$6$ to $4$,” or $6/4$. The order carries meaning: gold to blue is $4:6$, not $6:4$.
A ratio may compare part to part, part to whole, or whole to part. For the same box:
- blue to gold is $6:4$;
- blue to total is $6:10$;
- total to gold is $10:4$.
Name the quantities before writing numbers so the order remains clear.
Equivalent ratios
Multiplying or dividing both parts by the same nonzero number preserves the comparison:
This is the same equivalence principle used with fractions. A ratio in simplest whole-number form is useful, but an unsimplified ratio may reveal the original counts.
Ratio tables
A ratio table organizes equivalent pairs without hiding their meaning.
| Batches | Flour (cups) | Water (cups) |
|---|---|---|
| $1$ | $3$ | $2$ |
| $2$ | $6$ | $4$ |
| $5$ | $15$ | $10$ |
Each row preserves the flour-to-water ratio $3:2$. Scaling both entries together is essential.
Finding a missing quantity
You can also write a proportion:
Cross multiplication gives $5x=30$, so $x=6$. The ratio-table reasoning explains why that algebra works.
Dividing a total in a ratio
To divide $84$ in the ratio $3:4$, first count $3+4=7$ equal ratio parts. Each part is $84/7=12$. The two shares are $3(12)=36$ and $4(12)=48$.
Always check that the shares add back to the total: $36+48=84$.
Ratios, rates, and percent
A rate is a ratio comparing quantities with different units, such as kilometres to hours. A percent is a ratio to $100$. These ideas are related but the language matters: $3$ red counters for every $5$ blue counters is not automatically $3/5$ of all counters. Red is $3/(3+5)=3/8$ of the total.
Scale drawings and similarity
On a map with scale $1:50\,000$, one unit on the map represents $50\,000$ of the same units in reality. If two shapes are similar, corresponding side lengths share one constant scale factor. Lengths scale by that factor, areas by its square, and volumes by its cube.
Common mistakes
Reversing the order. Write labels such as blue:white above the numbers.
Scaling only one quantity. Equivalent ratios require the same multiplier or divisor on both parts.
Confusing part-to-part with part-to-whole. Add all parts before finding a fraction of the total.
Adding instead of multiplying. Moving from $2:3$ to $4:5$ adds $2$ to both entries but does not preserve the ratio.
Quick self-check
- What two quantities am I comparing, and in what order?
- Do they use the same units, or is this a rate?
- Did both quantities receive the same scale factor?
- If I divided a total, do the shares add back correctly?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Blue and white paint are mixed in the ratio 2:5. If 15 cups of white paint are used, how many cups of blue paint are needed?
- 15 ÷ 5 = 3, so the ratio is scaled by 3.
- 2 × 3 = 6.
- Use 6 cups of blue paint.
End of lesson
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